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Chapter 7 Local Validity, Grothendieck Topologies and Rough Sets 7.1 Representing Rough Sets The ﬁrst step is to represent rough sets. Thus, we now give the formal deﬁnition of a rough set and the formal deﬁnition of the decreasing representation of rough sets which was adopted in the Introduction. Deﬁnition 7.1.1. Given an Indiscernibility Space U, E, 1. Two sets X, Y ∈ ℘(U ) are called rough top equal, X 0 Y ,iﬀ (uE)(X)= (uE)(Y ). 2. Two sets X, Y ∈ ℘(U ) are called rough bottom equal, X∼Y ,iﬀ (lE)(X)= (lE)(Y ). 3. Two sets X, Y ∈ ℘(U ) are called rough equal, X ≈ Y ,iﬀ X 0 Y and X ∼ Y . 4. A set X ∈ ℘(U ) is called deﬁnable iﬀ X =(lE)(X)= (uE)(X). 5. A set X ∈ ℘(U ) is called undeﬁnable iﬀ (lE)(X)= ∅ and (uE)(X)= U . 6. Any equivalence class of subsets of U modulo the relation ≈ is called a rough set. 211 212 7 Local Validity, Grothendieck Topologies and Rough Sets We represent rough sets, with respect to an Indiscernibility Space U, E by means of ordered pairs of the form (uE)(X), (lE)(X),and
Published: Jan 1, 2008
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